Optimal Pheromone Utilization (in ANTS)

Optimal Pheromone UtilizationRoman KecherJoint work with: Yehuda Afek and Moshe Sulamy

Tel-Aviv UniversityAnts Nearby Treasure SearchInfinite gridk antsInitially at the originFood at distance D

Ants have to findthe food

Optimal run-time: (D + D2/k)[Feinerman, Korman, Lotker, Sereni, 2012]

WENS

Dk

(0,0)Ants are mobile agents essentially

Well be using Manhattan distance

PheromonesAnts emit pheromones[Lenzen, Radeva, 2013]Or notAnd sense them

No other communication

Biological resourceGoal: minimize pheromone count

WENS

Other communication models exist: constant size messages

Ants can not communicate in other forms, they cant even sense other ants

We would like to use minimal number of pheromones and still solve the problem effecientlyGround RulesEvery ant runs same algorithm (locally)With same initial state

Only uniform algorithms,ants have no knowledge of:k, total number of antsD, distance to the food

Synchronous ModelRounds:all ants move onceper round

WENS

Synchronous ModelRounds:all ants move onceper round

Assumption: antemission scheme[Emek, Langner, Uitto, Wattenhofer, 2013]At most one ant isemitted in each round

WENS

Note that there might be rounds where no ant is emitted; we only require constant (unknown) delays between emissionsAsynchronous Model

WENS

Adversary repeatedlyschedules one ant

Test&Set:Sense and emit a pheromone is oneatomic step

Definition of Rounds:Round ends when everyant took at least one stepOnly for (time) complexityAnts ModelsFSM: Finite State MachinesConstant size memory

TM: Turing MachinesUnlimited memory

Both deterministicResultsLower BoundAlgorithmFSM(Deterministic)TM(Deterministic)Previously known: O(D2) pheromones [Lenzen, Radeva, 2013]ResultsLower BoundAlgorithmFSM(Deterministic)(D) pheromonesto find the foodO(D) pheromonesO(D + D2/k) timeTM(Deterministic)Previously known: O(D2) pheromones [Lenzen, Radeva, 2013]ResultsLower BoundAlgorithmFSM(Deterministic)(D) pheromonesto find the foodO(D) pheromonesO(D + D2/k) timeTM(Deterministic)Results hold for Synchronous and Asynchronous modelsPreviously known: O(D2) pheromones [Lenzen, Radeva, 2013]Do not forget to say a word on Turing Machines, and that we only see half the story now, but will see the rest soonLayersDefinition: layer LAll grid cells at distance L from origin

WENSLThis is something we will need for the following slides

WENFSM Need (D) PheromonesAssume FSM withS statesUses o(D) pheromones

S+1 consecutivepheromone-free layers exist

Path starts and ends in same stateInfinite loop

SPath contains no pheromones - verballyFSM Algorithms

WENS

Problem:FSM cant count

Solution:Use pheromonesas turning points

Similar to the idea of guides[Emek, Langner, Uitto, Wattenhofer, 2013]

Note the duplicated walks on each cell

Basically, there are two problems:Enumerate the whole space this is shown herePrevent ants from doing duplicated work this is shown in the sync and async variantsAsynchronous FSM Algorithm

WENS

Mark E, S, W, N

Explore from NN never longer than E, S or W

Test&Set preventsmultiple ants fromexploring same layer

Note the duplicated walks on each cellSynchronous FSM Algorithm

WENSEmission schemebreaks initialsymmetry

But what happensif two ants collide?

Veteran ants behavedifferently thanNewbie ants

VeteranNewbie

This time we only explore every other layer (save duplication), to prevent misinterpretation of extra pheromoneResultsLower BoundAlgorithmFSM(Deterministic)(D) pheromonesto find the foodO(D) pheromonesO(D + D2/k) timeTM(Deterministic)Results hold for Synchronous and Asynchronous modelsPreviously known: O(D2) pheromones [Lenzen, Radeva, 2013]ResultsLower BoundAlgorithmFSM(Deterministic)(D) pheromonesto find the foodO(D) pheromonesO(D + D2/k) timeTM(Deterministic)(k) pheromonesfor optimal run timeO(k) pheromonesO(D + D2/k) timeResults hold for Synchronous and Asynchronous modelsPreviously known: O(D2) pheromones [Lenzen, Radeva, 2013]Async TM Need (k) Pheromones

WENSAssume oneant does not emitpheromones

Async TM Need (k) Pheromones

WENSAssume oneant does not emitpheromones

Consider samescheduling butwith extra antsAll new ants followthat one ant

Runtime remains thesame (but more ants)

Sync TM Need (k) Pheromones

WENSEmit one antUntil all pheromonesare placedEmit second antUntil all pheromonesare placedContinueDelay is constant

If no new pheromonesare placed, all followingants behave the same

Asynchronous TM Algorithm

WENSTM can count!Use pheromonesto assign IDs to ants

Static partitionExplore layersL = ID (mod Total)Occasionally updateestimated Total

Also works for thesynchronous model

ID = 1ID = 212111111122222222222222211111111111111111111111122222222222222222222222222222222

ThanksQuestions?